Incorrect p-values for nonparameteric statistical tests of Generalized Pareto Distribution

[Datseris]

Here is a MWE:

using Distributions, HypothesisTests

sigma = 1 / 2.0
xi = -0.1

gpd = GeneralizedPareto(0.0, sigma, xi)

X = rand(gpd, 10000)

TestType = OneSampleADTest
test = TestType(X, gpd)

p = pvalue(test)

fig, ax = hist(X; bins = 50, normalization = :pdf, label = "pvalue = $(round(p; digits=3))")
xrange = range(0, maximum(X); length = 100)
lines!(xrange, pdf.(gpd, xrange); color = :black, label = "analytic")
axislegend(ax)
fig 

Irrespectively of the parameters sigma, xi, and the RNG realization, the result is always very high p values. Instead the correct result would have been very low p values, because the data have very high confidence to come from the prescribed distribution. In the MWE the data are literally sampled by the distribution.

I've tried as hypothesis tests: OneSampleADTest, ApproximateOneSampleKSTest, ExactOneSampleKSTest. They all "fail" in the sense of not giving low enough p values.

[Datseris]

crosslinking https://discourse.julialang.org/t/testing-whether-data-come-from-a-generalized-pareto-distribution/102008 which shows that tthe tests fail also for a hand coded Cramer Von Mises test, so this may not be an issue with HypothjesisTests.jl...

[Datseris]

There is nothing wrong (see discord), I have simply misunderstood the test.